ardy and littlewood conjectured that every large integer $n$ that is not a square is the sum of a prime and a square. they believed that the number $mathcal{r}(n)$ of such representations for $n = p+m^2$ is asymptotically given by begin{equation*} mathcal{r}(n) sim frac{sqrt{n}}{log n}prod_{p=3}^{infty}left(1-frac{1}{p-1}left(frac{n}{p}right)right), end{equation*} where $p$ is a prime, $m$ is a...

The radiographic appearance of the petrotympanic fissure, through whic h the chorda tympani nerve exits the tympanic cavity, is familiar to radiologists as a normal struc ture, not to be mistaken for a frac ture of the anterior wall of the external auditory canal [1, 2). However, th e rad iographic demonstration of the canaliculus chordae tympan i, the bony canal which transmits the chorda tymp...

We study the problem of online weighted bipartite matching in which we want to find a maximum weighted matching between two sets of entities, e.g. matching impressions in online media to advertisers. Karp et al. [14] designed the elegant algorithm Ranking with competitive ratio 1−1/e for the unweighted case. Without the commonly accepted Free Disposal assumption, it is easy to show that no comp...

We show that there is a polynomial-time algorithm with approximation guarantee 3 2 + ε for the s-t-path TSP, for any fixed ε > 0. It is well known that Wolsey’s analysis of Christofides’ algorithm also works for the s-t-path TSP with its natural LP relaxation except for the narrow cuts (in which the LP solution has value less than two). A fixed optimum tour has either a single edge in a narrow ...

a proper vertex coloring of a simple graph is $k$-forested if the graph induced by the vertices of any two color classes is a forest with maximum degree less than $k$. a graph is $k$-forested $q$-choosable if for a given list of $q$ colors associated with each vertex $v$, there exists a $k$-forested coloring of $g$ such that each vertex receives a color from its own list. in this paper, we prov...